Kidder's spherical implosion law and the law for the implosion of a rotating cylinder

Kidder's law is obtained as follows: For an implosion velocity v _imp (where for ignition v _imp ~ 10⁸ cm/s), and M the mass of density ρ of an imploding spherical mass of radius R, the energy input E for ignition is

(1)$$E = \displaystyle{1 \over 2}{\rm MV}_{{\rm imp}}^2 \simeq {\rm \rho} R^3\simeq \displaystyle{{{\lpar {{\rm \rho} R} \rpar }^3} \over {{\rm \rho} ^2}}$$

for thermonuclear burn in DT one has ${\rm \rho} R \sim 1{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm g}/{\rm c}{\rm m}^2$, hence

(2)$$E\simeq \displaystyle{1 \over {{\rm \rho} ^2}}$$

but because

(3)$$\displaystyle{{\rm \rho} \over {{\rm \rho} _0}} = \left( {\displaystyle{{R_0} \over R}} \right)^3$$

where ρ and ρ₀ are the final and initial densities reached at R and R ₀ the final and initial radii. Hence

(4)$$E\simeq \left( {\displaystyle{R \over {R_0}}} \right)^6$$

or

(5)$$\displaystyle{{R_0} \over R}\simeq E^{-1/6}$$

which means that the ignition energy is very sensitive to the ratio R ₀/R, or vice versa, the ratio R ₀/R is very insensitive to the energy input and limited to the Rayleigh–Taylor instability.

Repeating the same analysis for a small cylinder of height h, one has

(6)$$E = \displaystyle{1 \over 2}Mv^2$$

or with M ≅ ρR ²h,

(7)$$E\simeq {\rm \rho} R^2hv^2$$

here we have

(8)$$\displaystyle{{\rm \rho} \over {{\rm \rho} _0}} = \left( {\displaystyle{{R_0} \over R}} \right)^2$$

but for a cylindrical implosion, one has

(9)$$\displaystyle{v \over {v_0}} = \displaystyle{{R_0} \over R}$$

hence with ρR ≅ 1 g/cm²

(10)$$E = \displaystyle{1 \over {\rm \rho }}hv^2 = \displaystyle{{hv_0^2 } \over {{\rm \rho }_0}}$$

which unlike (5) does not depend on R ₀/R.

The occurrence of the large negative mass densities in the general theory of relativity

While it is not possible to make a laboratory-size star, it is possible to reach a centrifugal field comparable to the gravitational field of a very dense star, with a density of the same order of magnitude as the density of a neutron star. It is the general theory of relativity (Landau and Lifshitz, Reference Landau and Lifshitz1951) in which the gravitational energy cannot be localized but is described by Einstein's energy-momentum pseudo tensor t^ik, where the energy and momentum are expressed by a sum of products of Christoffel symbols Γⁱ_kl, which are the forces. In a non-inertial reference frame, these forces are generally different from zero even in the absence of gravity-producing masses. As it was shown by Hund (Reference Hund1948), for non-relativistic velocities these forces can be obtained by Newtonian mechanics.

In the rotating reference frame inside the rapidly rotating target chamber, the equation of motion for a test particle is given by:

(11)$$m\displaystyle{{d{\bi v}} \over {dt}} = m{\bf F} + m\displaystyle{{\bi v} \over c} \times {\bf C}$$

where

(12)$${\bf F} = \; {\rm \omega} ^2{\bf r}$$

is the centrifugal force, and:

(13)$${\bf C} = \; -2{\rm c}{\bf \omega} $$

the Coriolis force, with ${\bf \omega} $ the angular velocity vector of rotation. If F is a gravitational acceleration produced by the mass distribution of density ρ as the source of Newton's law of gravity, one has:

(14)$${\rm div}{\bf F} = \; -4{\rm \pi} G{\rm \rho} $$

where G is Newton's constant. A centrifugal acceleration is also not free of sources, because of (12) one has

(15)$${\rm div}{\bf F} = 2{\rm \omega} ^2$$

and hence the negative gravitational field mass density

(16)$${\rm \rho} = \; -\displaystyle{{{\rm \omega} ^2} \over {2{\rm \pi} G}}$$

This negative mass is not fictitious and can be felt as the repulsive force in a merry-go-round. For ω = 0.6/s (an example given by Hund), one has ρ = −10⁶ g/cm³, comparable to the positive mass density of a white dwarf star.

The mass density (16) represents a physical reality as the mass density of the electric field E,

(17)$${\rm \rho} _{\rm e} = \displaystyle{{{\bf E}^2} \over {8{\rm \pi} c^2}}$$

While the mass density (17) is positive, because the equal sign charges repel each other, the mass density of a gravitational field g where equal sign masses attract each other, is negative and is

(18)$${\rm \rho} _{\rm g} = -\displaystyle{{{\bf g}^2} \over {8{\rm \pi} Gc^2}}$$

Likewise the mass density of the Coriolis field (13) is

(19)$${\rm \rho} = -\displaystyle{{{\bf C}^2} \over {8{\rm \pi} Gc^2}} = -\displaystyle{{{\rm \omega} ^2} \over {2{\rm \pi} G}}$$

equal to the mass density given by (16). This means the centrifugal force is caused by the negative mass density of the Coriolis force field. By comparison, the negative mass density $-\,{\bf F}^2/8{\rm \pi} Gc^2$ of the centrifugal force field (12) is smaller by the order v ²/c ².

Let us compare the negative gravitational field mass density with the positive mass density, ρ_N = 10¹⁴g/cm³, of a neutron star. With ω ≃ v/R ₀, where R ₀ ≃ 1 cm is the radius of the target chamber and $v\simeq 1{\kern 1pt} \; {\kern 1pt} {\rm km/s} = {\rm } 10^5{\rm cm/s}$, one finds that ${\rm \omega} = 2.6\; \, \times \,\,10^5{\kern 1pt} /{\rm s}$ and ρ = −10¹⁷ g/cm³, absolute three orders of magnitude larger than the mass density of a neutron star.

In the scientific literature little can be found about the fundamental importance of the Coriolis force in rotating plasmas. However, there are two references, one made by the author (Winterberg, Reference Winterberg1963), to an experimental verification of Elsasser's dynamo theory in rotating liquid metals, and the work by Haverkort and de Blank (Reference Haverkort and de Blank2012).

Nernst effect

In the proposed concept, the plasma is confined by the outwardly directed gravitational force of the negative mass of the Coriolis field, and by the inwardly directed magnetic force from the wall of the target chamber, by the Nernst effect.

With the temperature gradient $\nabla T$ from the cold wall into the hot plasma and B the magnetic field, the thermomagnetic current by the Nernst effect is for a hydrogen plasma given by Spitzer (Reference Spitzer1962):

(20)$${\bf j}_{\rm N} = \displaystyle{{3knc} \over {2B^2}}{\bf B} \times \nabla T$$

with the magnetic force density of the plasma:

(21)$${\bf f} = \displaystyle{1 \over c}{\bf j}_{\rm N} \times {\bf B} = \displaystyle{{3nk} \over {2B^2}}\lpar {{\bf B} \times \nabla T} \rpar \times {\bf B}$$

or with $\nabla T$ perpendicular and B parallel to the wall

(22)$${\bf f} = \displaystyle{3 \over 2}nk\nabla T$$

With the magnetohydrodynamic equilibrium condition

(23)$$\nabla p = {\bf f}$$

where $p = 2nkT,\; \nabla p = 2nk\nabla T + 2kT\nabla n$, one has

(24)$$2nk\nabla T + 2kT\nabla n = \displaystyle{3 \over 2}nk\nabla T$$

which upon integration yields

(25)$$Tn^4 = {\rm const}.$$

In a cartesian x, y, z coordinate system with the cold wall at z = 0, and the magnetic field B into the x-direction, the Nernst current density j is in the y-direction, and is

(26)$${\bf j}_y = -\displaystyle{{3knc} \over {2{\bf B}}}\displaystyle{{dT} \over {dz}}\; \; $$

From Maxwell's equations $4{\rm \pi} {\bf j}/{\rm c} = {\rm curl\;} {\bf B}$ one has

(27)$${\bf j}_y = \displaystyle{c \over {4\pi}} \displaystyle{{d{\bi B}} \over {dz}}$$

Eliminating ${\bf j}_y$ from (26) and (27) one obtains

(28)$$2{\bf B}\displaystyle{{d{\bf B}} \over {dz}} = {\bi \;} -12\pi kn\displaystyle{{dT} \over {dz}}$$

If in the plasma far away from the wall n = n ₀, T = T ₀, one has from (25)

(29)$$n = \displaystyle{{n_0T_0^{1/4}} \over {T^{1/4}}}$$

Inserting this into (28) one finds

(30)$$d{\bf B}^2 = -\displaystyle{{12\pi kn_0T_0^{1/4}} \over {T^{1/4}}}dT$$

With the boundary condition B = B ₀ at z = 0 and T = T ₀ at z = ∞, integration of (30) yields

(31)$$\displaystyle{{{\bf B}_0^2} \over {8\pi}} = 2n_0kT_0$$

The meaning of (31) is that the magnetic pressure ${\bf B}_0^2 /8{\rm \pi} $ exerted on the plasma from the wall surface at z = 0 balances the plasma pressure 2n ₀kT ₀ at z = ∞. With n ₀ = 10²³/cm³ and T ≃ 10⁸ K, one finds that B ₀ ≃ 10⁷ Gauss.

With the Nernst effect, where according to (25) Tn ⁴ = constant, the bremsstrahlung losses going in proportion to $n^2\sqrt T $ (Spitzer, Reference Spitzer1962) are constant across the entire plasma, which for nT = constant would near the wall rise in proportion to T ^−3/2. The reason are the large thermomagnetic currents set up in between the hot plasma and the cold wall of the target, leading to a large magnetic force repelling the hot plasma from the wall.

For the working of the Nernst effect one must have

(32)$${\rm \omega \tau} \gg 1$$

Where ω is the electron cyclotron frequency and τ is the electron–ion collision time. At high magnetic fields of the order B ≃ 10⁷ G and temperatures of the order 10⁸ K, with a particle number density n ≃ 10²³/cm³, one has ω ≃ 10¹⁴/s¹, τ ≃ 10⁻⁹s (Spitzer, Reference Spitzer1962) and hence ωτ ≃ 10⁵ ≫ 1.

Magnetohydrodynamic dynamo in the target chamber

A magnetohydrodynamic dynamo is ruled first by the equation (generalized Ohm's law in cgs units)

(33)$$\displaystyle{{\partial {\bf B}} \over {\partial t}} = \; \displaystyle{{c^2} \over {4{\rm \pi \sigma}}} \nabla ^2{\bf B} + {\rm curl}{\bi v} \times {\bf B}$$

where σ is the electrical conductivity, second, by the Euler equation of motion with the magnetic body force (1/c) j × B where j is the electric current density, neglecting viscous forces, and is given by

(34)$$\displaystyle{{\partial {\bf v}} \over {\partial t}} = \; -\displaystyle{1 \over {\rm \rho}} \nabla p-\displaystyle{1 \over 2}\nabla {\bf v}^2 + {\bf v} \times {\rm curl}{\bf v} + {\bi \;} \displaystyle{1 \over {{\rm \rho} c}}{\bf j} \times {\bf B}$$

in addition to the equation of continuity (mass conservation) and energy. For a uniform rotation, one can write

(35)$${\bf v} = {\bf \omega} \times {\bf r}$$

where ω = (1/2)curlv with (4π/c)j = curlB, one thus has for (34)

(36)$$\displaystyle{{\partial {\bf v}} \over {\partial t}} = \; -\displaystyle{1 \over {\rm \rho}} \nabla p + {\rm \omega} ^2{\bf r}\; -2{\rm \omega} \times {\bf v} + {\bi \;} \displaystyle{1 \over {4{\rm \pi \rho}}} {\bf B} \times {\rm curl}{\bf B}$$

Even without solving these equations, a general conclusion can already be drawn: For the buildup of the magnetic field by (33) one must have ∂B/∂t >0, which requires that the second term on the r.h.s of (33) is larger than the first term, or that the magnetic Reynolds number:

(37)$${\rm Rem} = \displaystyle{{4{\rm \pi \sigma} Rv} \over {c^2}} \gt 1$$

where $R\simeq 1{\kern 1pt} \,{\rm cm}$ is the radius of the target chamber and v the plasma velocity. Instead of (37) one can also write

(38)$$v \gt \; \displaystyle{{c^2} \over {4{\rm \pi \sigma} R}}$$

or that

(39)$${\rm \sigma} \gt \; \displaystyle{{c^2} \over {4{\rm \pi} Rc{\rm v}}}$$

Setting v = 10⁵ cm/s for the tangential velocity of the target chamber, one finds that σ ≳ 2.5 × 10¹³/s. On the other hand, the conductivity of a fully ionized plasma for T ≳ 10⁵ K is

(40)$${\rm \sigma} = 10^7T^{3/2}{\rm /s}$$

which means that for T ≳ 10⁵ K, one has σ ≳ 3 × 10¹⁴/s, or that for T ≳ 10⁵ K, σ is larger than the r.h.s. of (39).

With the plasma accelerated by the magnetic forces to a velocity of the order v ~ 10⁸ cm/s, (33) can be approximated extremely well by

(41)$$\displaystyle{{\partial {\bf B}} \over {\partial t}} = {\rm curl}\,{\bf v} \times {\bf B}$$

The magnetohydrodynamic instabilities arise from the last term on the r.h.s. of (36), at the moment when the magnetic forces overwhelm the fluid stagnation pressure or when B ²/8π ≥ (1/2)ρv ². For (1/2)ρv ² ≫ B ²/8π, the magnetic lines of force align themselves with the streamlines of the fluid flow. Then, if likewise the electric current flow lines j align themselves with ω, one has j orthogonal to B, and ω orthogonal to v. With ω = (1/2) curlv and j = (c/4π)curlB one can write for (1/2)ρv ² ≫ B ²/8π, that

(42)$$\vert {{\bf v} \times {\rm curl}{\bf v}} \vert \gg \; \left \vert {\displaystyle{{{\bf B}{\bi \;} \times {\rm curl}{\bf B}} \over {4{\rm \pi \rho}}}} \right \vert $$

or

(43)$${\bf v} \gt {\bf v}_{\rm A}$$

where

(44)$$\; v_{\rm A} = \displaystyle{B \over {\sqrt {4{\rm \pi \rho}}}} $$

is the Alfvén velocity. B will eventually reach the value $B = \sqrt {4{\rm \pi \rho}} v$, where v = v _A. In approaching this magnetic field strength, the magnetic pressure forces begin to distort the fluid flow which becomes unstable. In a plasma this leads to the formation of unstable pinch current discharges, where p = B ²/8π. For a hydrogen plasma of temperature T and particle number density n this leads to the Bennett equation

(45)$$B = \; \sqrt {16{\rm \pi} nkT} $$

The pinch instability can be seen as the breakdown of the plasma into electric current filaments, determined by the B × curlB term, whereas the v × curlv term is responsible for the breakdown of the plasma in vortex filaments. But while the breakdown into current filaments is unstable, the opposite is true for the breakdown into vortex filaments. This can be seen as follows: Outside a linear pinch discharge one has curlB = 0, and outside a linear vortex filament one has curlv = 0. Because of curlB = 0, the magnetic field strength gets larger with a decreasing distance from the center of curvature of magnetic field lines of force. For curlv = 0, the same is true for the velocity of a vortex line. But whereas in the pinch discharge a larger magnetic field means a larger magnetic pressure, a larger fluid velocity means a smaller pressure by virtue of Bernoulli's theorem. Therefore, whereas a pinch column is unstable with regard to its bending, the opposite is true for a line vortex. This suggests that a pinch column places itself inside vortex tube.

What is true for the m = 0 kink pinch instability is also true for the m = 0 sausage instability by the conservation of circulation

(46)$$Z = {\rm \oint} \; {\bf v}{\bi} \cdot {\bi} d{\bf r} = {\rm const}.$$

And because of the centrifugal force the vortex also stabilizes the plasma against the Rayleigh–Taylor instability.

In the proposed configuration, the plasma is radially confined between the convex centrifugal force in the target, and the concave centripetal magnetic force set up by the thermomagnetic current at the cold wall inside the target. According to Einstein's equivalence principle, the ions and electrons are affected by the centrifugal force in the same way as by the gravitational force in a star, having a stable configuration, but an instability may still arise in the thin boundary layer carrying the thermomagnetic current near the wall of the centrifuge. As Teller has pointed out in a private communication (Reference Teller1969), the instabilities of magnetic confinement configurations are likely to be absent in collision-dominated plasmas, or in plasmas where the mean free path λ is smaller than the linear dimension L of the magnetic confinement configuration. At the temperature T [K] and particle density n, the mean free path is likely given by

(47)$${\rm \lambda} = 10^4\; T^2/n$$

For n ≃ 10¹⁸/cm³ and T ≃ 10⁸ K, one has λ ≃ 10⁻² cm ≪ R ≃ 1 cm. This means that in this range of densities and temperature, the plasma is collision-stabilized.

In the past, rotating plasmas have been extensively studied by (Lehnert, Reference Lehnert1971), but only for densities where the plasma is not “collision dominated.” The proposed novel nuclear fusion concept is unique because it makes use of the self-exciting magnetohydrodynamic dynamo driven by the heat released from thermonuclear reactions in the fusion plasma. But it also has the potential to reach much larger magnetic fields for confinement and particle number densities than are otherwise possible.

This leaves open the question of how to remove heat from the target, even though this problem exists only for the 20% of fusion energy released in the target as charged particles, not for the 80% of the energy going into the kinetic energy of the neutrons, which can be slowed down outside the target over a much larger distance. One possible answer is to place the target in a supersonic potential gas vortex, for example, a vortex of helium gas, with the high velocity vortex core touching the outer surface of the target as a velocity of ~1 km/s, the tangential velocity of the target, with the vortex flow cooling the fusion target.

Ignition

The ignition is achieved by a <50 MJ pulse of an intense relativistic electron beam as it was originally proposed by the author in his 1968 paper published in the Physical Review (Winterberg, Reference Winterberg1968), except that here the beam stopping does not depend on the difficult to achieve two-stream instability, because for MeV electrons the stopping range is determined in the ferromagnetic mantle of the target, which implodes the target. This stopping range is given by (Winterberg, Reference Winterberg2010):

(48)$${\rm \lambda} \simeq \displaystyle{1 \over {\rm \rho}} \lpar {0.543\,E-0.16} \rpar \,\,{\rm cm}$$

Where ρ is the target density in g/cm³, and E is the electron energy in MeV. For ρ ≃ 10 g/cm³ and E ≃ 10 MeV, one obtains λ ≃ 1 cm, about equal to the linear dimension of the target.

With 10 MJ ≃ 10¹⁴ erg, deposited in a volume $V\simeq {\rm \pi} R_0^2 h \gt 1\,{\kern 1pt} {\rm c}{\rm m}^3$, the temperature given by E ≃ nkT, with n ≃ 10²³ particles/cm³, is T ≃ 10⁶ K. The threefold implosion of the DT from $R\simeq h\simeq 1{\kern 1pt} $ to $R\simeq h\simeq 0.3{\kern 1pt} \,{\rm cm}$ increases its temperature by the factor of 100, from T ≃ 10⁶ to T ≃ 10⁸ K, meaning T would reach the ignition temperature of the DT reaction at T ≃ 10⁸ K.